Center (snap) hole in object
You can do this in 3 stages, that do involve the cursor but in a different manner:


Select the outer border vertices of your door and snap you cursor to this selection (ShiftS > Cursor to Selected). This will move the cursor to the center of your door object.

Now, select the geometry delimiting your hole.
Open the Properties Panel (N), and copy the XYZ values from the 3D Cursor location to the Median (which represents the center location of your selection). This will move your selection to the cursor, while maintaining the shape of the hole.

EDIT: Or instead of copying values, you can use the snap tool to move the selection to the cursor (ShiftS > Snap Selection To Cursor (Offset). The offset option makes sure the selection moves while maintaining its shape (thanks @Mr Zak!).


Result: